Dissipativity and Gevrey regularity of a Smoluchowski equation

Constantin, Peter; Titi, Edriss S.; Vukadinovic, Jesenko

doi:10.1512/iumj.2005.54.2653

Cited by 25 publications

(22 citation statements)

References 13 publications

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“…We can show, as claimed in [3], [4] that at any time t > 0 the solution is analytic in the space variable. The idea is to show, following [4] (based on [2], [10]), that the solution is in some Gevrey class of functions, defined by a parameter depending on time. This class is a subset of the set of real analytic functions on the sphere.…”

Section: Together With Poincaré Inequality Solving This Inequality Gsupporting

confidence: 76%

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Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Frouvelle¹,

Liu²

2012

SIAM J. Math. Anal.

119

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Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing else than the Doi equation (also called Smoluchowski equation) with dipolar potential.In a self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady-states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, there is also a family of non-isotropic equilibria.We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.

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Section: Together With Poincaré Inequality Solving This Inequality Gsupporting

confidence: 76%

“…Following [4], we will show that the solution belongs to a special Gevrey class. We define the space G r , as the set of functions g (with mean zero) such that ∆…”

Section: A Appendixmentioning

confidence: 99%

Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Frouvelle¹,

Liu²

2012

SIAM J. Math. Anal.

119

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“…Existence and uniqueness of solutions for (1.1) in L 2 or Sobolev H s spaces is a classical result. Combining a result of [4] and a method of [7], we show that for any initial condition q 0 in a measure class the (unique) solution q(t) of (1.1) is in a analytical functions space for all times t > 0. One consequence of the regularizing properties of equation (1.1) is that many convergence phenomena of solutions towards equilibria or invariant manifolds happen in an analytic-functions space and not just in the classical L 2 space.…”

Section: Introductionmentioning

confidence: 84%

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

2012

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Abstract. We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Otherwise, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disk composed of radial trajectories connecting a saddle-point equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and coherent (or synchronized) equilibria. We prove in particular non-linear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.2010 Mathematics Subject Classification: 37N25, 37B25, 92B25, 82C26

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“…For example, in [9] it has been shown that bounds on the analyticity radius can be employed to determine whether a given initial data lies near the global attractor. The technique introduced in [10] has been extended to the case of certain nonlinear parabolic equations in [7], and, very recently, to the Smoluchowski equation in [6]. The study of the space analyticity radius for the NSE and KSE in the setting of a Banach (L p ) space was first carried out by Grujić and Kukavica [16].…”

Section: The Kuramoto-sivashinsky Equation (Kse) Ismentioning

confidence: 99%

Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in Rn

Biswas

Swanson

2007

Journal of Differential Equations

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Motivated by the work of Foias and Temam [C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989) 359-369], we prove the existence and Gevrey regularity of local solutions to the Kuramoto-Sivashinsky equation in R n with initial data in the space of distributions. The control on the Gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data. In the particular case when n = 1, our analysis allows for initial data that are less smooth than that considered by Grujić and Kukavica [Z. Grujić, I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in L p , J. Funct. Anal. 152 (1998) 447-466].

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Dissipativity and Gevrey regularity of a Smoluchowski equation

Abstract: We investigate a Smoluchowski equation (a nonlinear FokkerPlanck equation on the unit sphere), which arises in modeling of colloidal suspensions. We prove the dissipativity of the equation in 2D and 3D, in certain Gevrey classes of analytic functions.

Cited by 25 publications

References 13 publications

Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in Rn

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